Enriched monoidal categories I: centers

Understand

This work is the first one in a series, in which we develop a mathematical theory of enriched (braided) monoidal categories and their representations.

• In this work, we introduce the notion of the $E_0$-center ($E_1$-center or $E_2$-center) of an enriched (monoidal or braided monoidal) category, and compute the centers explicitly when the enriched (braided monoidal or monoidal) categories are obtained from the canonical constructions.
• These centers have important applications in the mathematical theory of gapless boundaries of 2+1D topological orders and that of topological phase transitions in physics.
• They also play very important roles in the higher representation theory, which is the focus of the second work in the series.